Theorem pwundif 5021

Description: Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)

Assertion

Ref	Expression
pwundif	|- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Proof of Theorem pwundif

Step	Hyp	Ref	Expression
1		undif1 4043	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) = ( ~P ( A u. B ) u. ~P A )
2		pwunss 5019	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3787	. . . . 5 |- ( ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) ) <-> ( ~P A u. ~P B ) C_ ~P ( A u. B ) )
4	2, 3	mpbir 221	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 474	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3786	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 220	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	eqtr2i 2645	1 |- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Syntax hints:   /\ wa 384   = wceq 1483   \ cdif 3571   u. cun 3572   C_ wss 3574   ~Pcpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160

This theorem is referenced by:  pwfilem  8260